Stream classic is the experience on web.microsoftstream.com
Stream on SharePoint is a fancy way to say videos uploaded to SharePoint, OneDrive, Teams, Yammer, or other M365 service using SharePoint file store.
The license your see to Stream for your users is stream classic. For Stream on SharePoint we aren't using a separate liscense it's just the normal SharePoint liscense.
We have a new SharePoint admin setting on the roadmap starting roll out in March I believe, that will let you let you change the tile in the app launcher to point Stream at stream.office.com (our new get back to videos experience) instead of Stream classic.